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Elkin, Yu.E.; Biktashev, V. N.

doi:10.1023/a:1005187225866

Cited by 11 publications

(7 citation statements)

References 24 publications

(50 reference statements)

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“…As stated in [52], the prevalence of multiple orbits may be understood in terms of asymptotic theories involving further small parameters. So, the version of kinematic theory of spiral waves suggested in [56] produces an equivalent of response functions, which is not only quickly decaying, but also periodically changing sign at large radii, with an asymptotic period equal to the quarter of the asymptotic wavelength of the spiral wave. Other variants of the kinematic theory, e.g.…”

Section: Discussionmentioning

confidence: 99%

Computation of the drift velocity of spiral waves using response functions

et al. 2010

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Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. In the presence of a small perturbation, the spiral wave's center of rotation and fiducial phase may change over time, i.e., the spiral wave drifts. In linear approximation, the velocity of the drift is proportional to the convolution of the perturbation with the spiral's response functions, which are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues λ=0,±iω . Here, we demonstrate that the response functions give quantitatively accurate prediction of the drift velocities due to a variety of perturbations: a time dependent, periodic perturbation (inducing resonant drift); a rotational symmetry-breaking perturbation (inducing electrophoretic drift); and a translational symmetry-breaking perturbation (inhomogeneity induced drift) including drift due to a gradient, stepwise, and localized inhomogeneity. We predict the drift velocities using the response functions in FitzHugh-Nagumo and Barkley models, and compare them with the velocities obtained in direct numerical simulations. In all cases good quantitative agreement is demonstrated.

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Section: Discussionmentioning

confidence: 99%

Computation of the drift velocity of spiral waves using response functions

et al. 2010

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“…Spiral wave drift due to media inhomogeneities is well known: it was studied in numerical simulations [39] and then in the experiments with the heart tissue [40,41] and in the Belousov-Zhabotinsky reaction [42]. Attempts to explain or predict the direction and velocity of the drift have been made [39,[43][44][45], but they were based on various phenomenological arguments applicable to narrow classes of autowave media with special properties, while the response functions method allows to predict the spiral wave drift velocity due to weak media inhomogeneities without any restrictions on the type of inhomogeneity.…”

Section: Drift Of Spiral Wavesmentioning

confidence: 99%

Wave-particle dualism of spiral waves dynamics

Biktasheva

Biktashev

2003

Phys. Rev. E

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We demonstrate and explain a wave-particle dualism of such classical macroscopic phenomena as spiral waves in active media. That means although spiral waves appear as nonlocal processes involving the whole medium, they respond to small perturbations as effectively localized entities. The dualism appears as an emergent property of a nonlinear field and is mathematically expressed in terms of the spiral waves response functions, which are essentially nonzero only in the vicinity of the spiral wave core. Knowledge of the response functions allows quantitatively accurate prediction of the spiral wave drift due to small perturbations of any nature, which makes them as fundamental characteristics for spiral waves as mass is for the condensed matter.

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“…The possibility of orbital drift, related to a change of sign of an equivalent of F r (d) was discussed at a speculative level in [22]; how often this phenomenon may occur in reality is a more complicated question. The equivalent of response functions calculated in [23] has a structure which suggests that for large-core spirals there is an infinite set of orbits. In practice, orbital motion can only be observed for lower orbits where the orbiting speed is noticeable.…”

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confidence: 99%

Orbital Motion of Spiral Waves in Excitable Media

2010

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Spiral waves in active media react to small perturbations as particlelike objects. Here we apply asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation center along circular orbits around the inhomogeneity. The stationary orbits have fixed radii and alternating stability, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct numerical simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment. [7] biology). In a perfectly uniform medium the core of a spiral wave may be anywhere, depending on initial conditions. However, real systems are always heterogeneous, and therefore spiral drift due to inhomogeneity is of great practical interest to applications. Understandably, such drift has been mostly studied in excitable chemical reactions and the heart, where drift due to a gradient of medium properties [8,9] and pinning [10] (anchoring, trapping) to a localized inhomogeneity [11][12][13] have been observed in experiments and simulations. Interaction with localized inhomogeneity can be considered to be a particular case of the general phenomenon of vortex pinning to material defects [14].Here we identify a new type of spiral wave dynamics: precession around a localized inhomogeneity along a stable circular orbit. We predict this novel phenomenon theoretically, describe its key features, and confirm it by numerical simulations. We argue that this orbital motion of spiral waves is robust and prevalent, has nontrivial and surprising consequences for applications, and should be directly observable in experiments.We consider reaction-diffusion equations, which is the most popular class of models describing spiral waves:where u, f 2 R ' , D 2 R 'Â' , p 2 R m , uðr; tÞ is the dynamic vector field,r 2 R 2 , pðrÞ ¼ p 0 þ p 1 ðrÞ, jp 1 j ( 1, is the vector of parameters, D is diffusion matrix. For p ¼ (1) is assumed to have spiral wave solutions rotating with angular velocity ! (taken here to be clockwise for ! > 0),where ( , #) are polar coordinates defined with respect to the center of rotationR ¼ ðX; YÞ T , and È is the initial rotation phase.In the presence of a small perturbation p 1 ðrÞ Þ 0, the spiral's center of rotation R ¼ X þ iY is not constant but slowly evolves with the equation of motionwhere ¼ ðr ÀRÞ and ¼ #ðr ÀRÞ þ ! À È are polar coordinates in the corotating frame of reference, and h is the perturbation to the right-hand side of Eq. (1

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Abstract: Spiral waves in excitable media may drift due to interaction with medium inhomogeneities. We describe this drift asymptotically, within the kinematic (eikonal) approximation.

Cited by 11 publications

References 24 publications

Computation of the drift velocity of spiral waves using response functions

Computation of the drift velocity of spiral waves using response functions

Wave-particle dualism of spiral waves dynamics

Orbital Motion of Spiral Waves in Excitable Media

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